Aircraft: COMMERCIAL FIXED WING
Phase of flight: Cruise
Anomaly: ATC Issue All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy

We departed at XA15 Zulu bound for Europe. There are no NavTracks at this time of day east bound so we were on a random route. I was the IRO (Relief pilot) and on break when the oceanic clearance was received. The clearance was a reroute.I returned to the flight deck at XC35 Zulu and was getting settled in when we crossed N50W050. Minutes after crossing 50W Gander messaged us to call on HF. When we checked in with Gander he asked for our north coordinate at 30W. We both checked the FMS and then decided to verify against the printed clearance. The FMS was incorrect. We verified with Gander that our reroute clearance and coordinates were correct and changed the FMS to reflect the correct clearance. Our 40W coordinate were the same (original route and reroute) so we never turned off course.All three crew members then silently verified the clearance (the 3rd was recalled to the flight deck). Then the pilot monitoring (PM) read the coordinates out loud to the pilot flying and the FMS coordinates were confirmed. No further action was needed. We never turned off course and landed safely at destination. We received our random route oceanic clearance at about the same time that the ride deteriorated at our flight-planned crossing altitude. We climbed to FL390 and asked Moncton to forward a request for Gander to re-clear us to cross at the higher altitude. He complied and we received that clearance. We followed SOP in the route confirmation process and everything looked normal. We were given direct to 50N050W and proceeded accordingly. Following the crew change and soon after passing 50N050W and now navigating to 51N040W; we received a CPDLC message from Gander Radio to contact them on 8831; which we did. He told us he couldn't read our north coordinate at 30W on the position report and to please confirm what our clearance showed. We checked and found that we still had our originally filed points of 50N030W and 50N020W in the FMC and not the cleared points issued to us of 51N030W and 51N020W. We changed the points and sent the info to dispatch. The leg we were on (to 51N040W) was correct so a Gross Navigation Error didn't happen but we unintentionally set ourselves up for one. I don't know if the Gander Radio operator couldn't actually read the report or if that was a tactful way of saying 'Check your planned flight route because what I see doesn't match.' The most ironic thing about this almost-incident was that the Captain and I had quite literally just finished re-reading the Atlantic oceanic route clearance procedures in the FOM. In spite of that; we still managed to miss digit changes on two points! We apparently saw-and/or-read what we expected to see instead of what was actually there.As a side note; we did not receive the normal required message from ATC stating that the clearance we received was different from the filed route. We think this might have happened because the Moncton controller relayed the message concerning our ALTITUDE change and he had no knowledge of the oceanic ROUTE change so he didn't say anything. We think the Gander controller probably believed that we were verbally informed of the route change at the same time that we received the altitude change. Even though we didn't receive the verbal; we still had a piece of paper in the cockpit that did have the changes and we missed them. An air crew received a revised random route clearance prior to entering oceanic airspace. They did not revise the route in the FMS. Fortunately they checked their routing after Gander Radio asked them to confirm the waypoint after the one toward which they were navigating. They revised the routing in the FMS and were never off course.